Theorem nnssn0s 28078

Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
nnssn0s	⊢ ℕs ⊆ ℕ0s

Proof of Theorem nnssn0s

Step	Hyp	Ref	Expression
1		df-nns 28073	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
2		difss 4131	. 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s
3	1, 2	eqsstri 4016	1 ⊢ ℕs ⊆ ℕ0s

Syntax hints:   ∖ cdif 3945   ⊆ wss 3948  {csn 4628   0_s c0s 27669  ℕ_0scnn0s 28070  ℕ_scnns 28071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-dif 3951  df-in 3955  df-ss 3965  df-nns 28073

This theorem is referenced by:  nnssno  28079  nnsgt0  28092